For \( \alpha =1 \)  the above results are standard (cf. [1]}). Hence,
by the method of transposition (cf. [2]}), we define the very weak
solution \( y \in L^2(0,T;Y) \)  to the state equation of problem eq:model
with \( u \in L^2(0,T;Z) \)  by that
\(\int _0^T \big (y, \, ({T-}^\alpha - \mathcal {A}) \varphi \big )_Y \, \mathrm {d}t =-\langle {u, \partial _{\mathbf {n}} \varphi } \rangle _{\partial \Omega \times (0,T)}\) 
